RH Toolkit 3 — Divisor Geometry and Arithmetic Curvature

This paper develops a discrete operator-theoretic framework for studying arithmetic curvature associated with the divisor function. It defines curvature, oscillation, and baseline-deviation components using finite-difference operators and subtraction of the classical logarithmic average baseline B(n) = log n + (2 gamma - 1). These components are assembled into an arithmetic curvature vector and a weighted curvature energy functional on polynomially decaying weighted l2 spaces. Using only standard Hilbert-space operator theory and the classical estimate d(n) = O(n^epsilon) for every epsilon > 0, the paper proves boundedness, adjointness, weighted summability, and finiteness of the resulting energy under explicit decay conditions. The framework also records compatibility with translation-invariant operators, Fourier multiplier representations, sign-modulation commutators, localized perturbations, compact energy operators, and symmetry-sector decompositions. The appendices extend the main construction in three directions. First, they provide a finite-dimensional matrix realization of the curvature framework suitable for computation on finite windows. Second, they formulate a transport-and-extraction architecture in which arithmetic residuals are represented as localized geometric operators up to compact or Hilbert-Schmidt correction. Third, the new final appendix adds a structural admissibility filter for xi-induced residual realizations: in the canonical transport-and-extraction framework, admissible residuals must lie in a geometric-plus-Hilbert-Schmidt class, consisting of a bounded variable-coefficient finite-band geometric operator together with a Hilbert-Schmidt remainder. This final addition supplies a class-membership and detector-completeness interface that can be used for research frameworks. Together, the paper provides a modular interface connecting divisor-function data, discrete differential operators, weighted energy functionals, localized residual geometry, and operator-theoretic stability principles.